13 October 2012

On learning and Bayesian statistics

I have--rather to my surprise--just finished reading Sharon Bertsch McGrayne's The theory that would not die (London; Yale U P, 2011). I only picked it up because there was a 3 for 2 offer in the bookshop and I needed a third book in order not to miss out. The sub-title says it all: "How Bayes' rule cracked the Enigma code, hunted down Russian submarines, and emerged triumphant from two centuries of controversy." I first came across Bayes' theorem through an Open University course on Professional Judgment* in the later '80s, in the days when their courses were supported by late-night programmes on TV. As usual with those programmes, particularly in maths, I thought I understood them perfectly at the time but the understanding just evaporated within hours... Even so, the basic idea has stuck with me for twenty-odd years, which is more than I can say for the MOOC...

So what is Bayes/Laplace's theorem/rule? My understanding is limited, but it is fundamentally a statistical approach to informing decision-making under conditions of uncertainty and incomplete information, by refining probabilities based on information about prior conditions as well as current measurements/ assessments. To me, the overall process seems very like the way in which betting odds are arrived at (where the money bet is a proxy for positive or negative information about the object of the wager); or as Keynes put it, "When the facts change, I change my opinion. What do you do, sir?" (epigraph to McGrayne).

Here is a more respectable, and of course more detailed, account.

I was already aware of some of the basic principles as applied to assessment (here)...But here is why I am mentioning it now--it came up as a current story. 

McGrayne doesn't make the links to learning very much (just on p.158 and pp.247/8.) But it's not a giant leap of insight to make the connection between learning through successively better approximations and Bayesian principles:
  • Piagetian principles of adaptation-- assimilation and accommodation--are consistent with a Bayesian framework.The progress of a Bayesian calculation--up-dating predictions or odds on the basis of greater knowledge, is basically similar to Piaget's view of children's learning through assimilation, where each new instance of a concept informs and adds to existing understanding (as discussed in the bold linked article above).
  • And Hattie's meta-analytic findings on the importance of feedback  also fit, particularly the passage I quote on feedback from learner to teacher. The teacher is continually making adjustments to her strategy based on refinement of her uncertain and imperfect knowledge--some Bayesian understanding of the process may well help with the formulation of her strategy, and although this may be difficult to achieve in real time in the classroom, one can see how it could possibly be incorporated into an assessment strategy, or indeed an online course.
Or is the characteristic sound of a Sunday morning the sound of me barking up the wrong tree? (As Penelope Gilliat said of Harold Hobson.) The probability of that could also be computed using Bayes'. Probably.

*  See the reader for the course: Dowie J and Elstein A (eds.) (1988) Professional Judgment; a reader in clinical decision making Cambridge; Cambridge U P. Incidentally I also wrote about it here. What I realise only now--since reading McGrayne-- is that the dissenting voice granted a couple of minutes at the end of each programme represented an almighty row which was going on in the world of statistics; the main programme was unashamedly Bayesian, while the comment presumably represented the mainstream view, which McGrayne typifies as "frequentist".

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